The auxiliary police, who provide a back-up to the regular Watch at times of great need. A typical time of great need, is described in Men at Arms, where a young Watchman named Carrot calls up the citizenry to save Ankh-Morpork from itself.
Specials include Andy Hancock, a very enthusiastic clacksman, and government inspector A.E. Pessimal, a man born to bureaucracy and not so much named as initialed. He discovered the Watchman's life was for him when the pent-up aggression of years suddenly released itself in a berserk attack on an astonished troll. The president of the Thieves' Guild, Mr Boggis, is also a very unlikely Special, as are his bodyguards Vinny "No Ears" Ludd (Ludd is a common surname given to foundlings taken in by the Guild) and Harry "Can't remember his own nickname" Jones. As this adds extra professional muscle to the muster, Vimes is prepared to be accepting. The ranks of Specials still include the Librarian (of Unseen University), whom Nobby called "Special Ape Services" (SAS?) in Guards! Guards!.1 Vimes's butler Willikins, a man of hidden depths as well as of fish-hooks sewn into his lapels and sharpened coins hidden in the brim of his cap, is also a Special.
Not everyone is taken on as a Special: the Dean of Unseen University was thanked very much for his interest and sent home. Carrot was undoubtedly aware of Vimes' suspicion of magic as a police tool, as well as understanding that the Dean's default position of zapping everything with a blast from the good ol' staff might prove counter-productive.
The New Discworld Companion also hints that there is a different, unadvertised, invitation-only, category of Special: the new, reformed Cable Street Particulars would appear to be brought together on an ad-hoc, as-needed, basis, composed of people with other full-time jobs in the City, but whose particular professional skills and aptitudes are ideal for solving a different sort of crime, the sort whose prosecution requires intellect and the slow, patient, acquisition and sifting of available information, so as to build up a picture of those crimes which aren't easily seen and which take care not to advertise themselves. It is possible that the pianist and organist at the Opera House is such a special, combining police work with his full-time passion for popular music. (Oddly enough, Roundworld has such a fictional detective, the cerebral and musicologically inclined Inspector Morse, who plies his trade, happily for a TV company looking for overseas sales, in and around the photogenic old English city of Oxford.)
1 As for the Librarian, who has many resources at his fingertips, it could be said that this is a "special branch" for him to swing from.